In chapter 1 of his Utilitarianism, John Stuart Mill puts fourth an argument in favor of the “Greatest Happiness Principle” or GHP. That is, he argues that a thing is good insofar as it produces the greatest happiness for the largest number of people. While his ideas may seem at first to be an intuitive extension of the Hedonistic Value Theory, when examined carefully through logical analysis, it can be shown to be not only unsound, but actually totally invalid as an argument.

The argument follows like this:

1. If everyone desires his own happiness, then individual happiness is desirable. (Premise.)
2. Everyone desires his own happiness. (Premise.)
3. If everyone’s individual happiness is desirable, then the happiness of everyone is desirable. (Premise.)
4. Every individual’s happiness is desirable. (By conditional elimination from 1 and 2)
5. Therefore the happiness of everyone is desirable. (By conditional elimination from 3 and 4)
6. If the happiness of everyone is desirable, then an action is right, if and only if it produces the greatest happiness for the greatest number of people. (Premise.)
7. Therefore, an action is right, if and only if is produces the greatest happiness for the greatest number of people. (By conditional elimination from 5 and 6.)

While this argument seems to flow logically, it encounters a problem in the fact that the argument itself is actually invalid. Assuming this is the argument made by Mill, the argument is invalid because it relies on the Fallacy of Equivocation. Mill falsely conflates two different variations on the meaning of desirable, making appear as if something being “capable of being desired” also means it should be desired. The use of the world desirable to refer to the former in the first several premises of the argument, followed by the use of the latter in the conclusion, makes the argument inconsistent, and by consequence, invalid.

In order to make this argument function, one would need to eliminate the false equivocation of the two forms of “desired” by the addition of a premise. Changing the word “desirable” to “capable of being desired” and “ought to be desired” to draw distinction between the two variations would clarify the meaning of the argument, and the addition of the premise “If something is capable of being desired, then it ought to be desired” would add to the validity of the argument. This argument would still be unsound, however, due to the fact that this additional premise is not necessarily true. It seems the problems with the premises and the reliance on fallacies are difficult to overcome in this argument.

Mill’s argument, taken in this way, is both unsound and invalid, and it is difficult to make an argument along the same lines which doesn’t fall into the same issues. The basis of the GHP as a useful ethical principle is quite flawed, and its use in practice is not philosophically viable. This doesn’t necessarily eliminate Utilitarianism as a potentially valid ethical theory, it only draws into question the value of J.S. Mill’s Utilitarianism and his argument for it.